where the minimum is obtained by the dephased state \(\Delta [\rho _b(t)]=\sum _n\langle \varepsilon _n|\rho _b(t)|\varepsilon _n\rangle |\varepsilon _n\rangle \langle \varepsilon _n|\) .
For pure states, the Von Neumann entropy \(S(t)=-\mathrm {Tr}[\rho _b(t)\log (\rho _b(t))]\) 
characterizes the battery-charger entanglement [1]} and thus the diagonal entropy \(S_\Delta (t)=-\mathrm {Tr}[\Delta [\rho _b(t)]\log (\Delta [\rho _b(t)])]\)  quantifies the sum of entanglement and coherence.
